Every result that can happen
Toss a coin and it lands heads or tails. Each possible result is called an outcome. A fair coin has two outcomes, heads (H) and tails (T), and they are equally likely.
The list of every outcome is called the sample space. For a coin, the sample space is H, T.
A fair coin has two equally likely outcomes, so each one takes half of the circle.
List them, then count
A fair dice has six outcomes: 1, 2, 3, 4, 5 and 6. Once the sample space is written out, a question about the dice becomes counting.
How many of the outcomes are even numbers? Three of them: 2, 4 and 6. How many are more than 4? Two of them: 5 and 6.
The sample space of a dice, with the 3 even outcomes colored: 2, 4 and 6.
Two things at once
When two things happen together, each outcome is a pair. Toss a coin and roll a dice, and one outcome is heads with a 3, written H3.
A table keeps the pairs in order: one row for each side of the coin, and one column for each face of the dice. Every cell is one outcome. There are 2 rows of 6 cells, so there are 2 × 6 = 12 outcomes.
The rows are the coin and the columns are the dice. 2 rows of 6 make 12 outcomes, from H1 to T6.
Two dice make 36
Roll two dice, a red one and a blue one. The red dice has 6 outcomes, and for each of them the blue dice has 6, so there are 6 × 6 = 36 outcomes.
In the grid, the rows are the red dice and the columns are the blue dice. Each cell shows the total of the two numbers. Six rows of six can be counted, where a list of 36 pairs would have to be trusted.
Each cell is one outcome of the two dice, with the total of the two numbers written in it.
A 3 and a 4 is not a 4 and a 3
The usual mistake is to list only the totals. Two dice can total anything from 2 to 12, which is 11 different totals, but there are 36 outcomes, not 11. A 3 on the red dice with a 4 on the blue dice is a different outcome from a 4 on the red dice with a 3 on the blue dice, even though both total 7.
The totals are not equally likely, either. Six cells total 7, and only one cell totals 2: a 1 on both dice. The grid shows this at a glance, and a list of 11 totals hides it.
The 6 outcomes that total 7 lie on a diagonal: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, and 6 and 1.
Worked example: A Coin and a Three-Color Spinner That Decide Who Moves First
Question To start a board game, Ana tosses a fair coin and spins a spinner with three equal sections colored red, blue and yellow. (a) List all the possible results, such as heads and red. How many results are there? (b) Ana moves first if the coin shows heads, or the spinner shows red, or both. What is the probability that Ana moves first?
1.Draw a table. The rows are heads (H) and tails (T). The columns are red, blue and yellow.
The rows are the two sides of the coin. The columns are the three colors. 2.(a) Each cell is one result: H and red, H and blue, H and yellow, T and red, T and blue, T and yellow. There are 2 × 3 = 6 results, and they are equally likely.
(a) Each cell is one result: 2 × 3 = 6 equally likely results. 3.Shade the cells where Ana moves first: all 3 cells in the heads row, and the tails cell under red. The cell H and red is shaded once, not twice.
Shade the heads row and the tails cell under red. H and red is shaded once. 4.(b) 4 of the 6 cells are shaded, so the probability that Ana moves first is 46 = 23.
(b) 4 of the 6 cells are shaded: 46 = 23.
Answer: (a) 6 results; (b) 23
Common mistakes
- Adding 12 for heads and 13 for red to get 56. The result H and red is in both groups, so it is counted twice. Count the shaded cells instead.
- Listing only 5 results: heads, tails, red, blue and yellow. Each result is a pair, one side of the coin with one color, so there are 2 × 3 = 6 results.